Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $p = \dfrac{-t + 8}{t^2 - 12t + 32} \times \dfrac{-t + 4}{t + 8} $
Answer: First factor the quadratic. $p = \dfrac{-t + 8}{(t - 4)(t - 8)} \times \dfrac{-t + 4}{t + 8} $ Then factor out any other terms. $p = \dfrac{-(t - 8)}{(t - 4)(t - 8)} \times \dfrac{-(t - 4)}{t + 8} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ -(t - 8) \times -(t - 4) } { (t - 4)(t - 8) \times (t + 8) } $ $p = \dfrac{ (t - 8)(t - 4)}{ (t - 4)(t - 8)(t + 8)} $ Notice that $(t - 8)$ and $(t - 4)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ (t - 8)\cancel{(t - 4)}}{ \cancel{(t - 4)}(t - 8)(t + 8)} $ We are dividing by $t - 4$ , so $t - 4 \neq 0$ Therefore, $t \neq 4$ $p = \dfrac{ \cancel{(t - 8)}\cancel{(t - 4)}}{ \cancel{(t - 4)}\cancel{(t - 8)}(t + 8)} $ We are dividing by $t - 8$ , so $t - 8 \neq 0$ Therefore, $t \neq 8$ $p = \dfrac{1}{t + 8} ; \space t \neq 4 ; \space t \neq 8 $